Introduction:
When a small steel mass is whirled in a horizontal circle with a uniform angular velocity, and the string is suddenly pulled so that the radius of the circle is halved, the new angular velocity of the steel mass will change. To understand this change, we need to consider the conservation of angular momentum.

Conservation of Angular Momentum:
Angular momentum is conserved when no external torque acts on a system. The angular momentum (L) of an object is given by the product of its moment of inertia (I) and its angular velocity (ω): L = Iω.

Initial Situation:
In the initial situation, the small steel mass is whirled in a horizontal circle with a uniform angular velocity of 2ω. The moment of inertia (I) of the steel mass remains constant.

Change in Radius:
When the string is suddenly pulled so that the radius of the circle is halved, the moment of inertia (I) of the steel mass changes. The moment of inertia of a point mass about an axis passing through its center of mass is given by the formula: I = MR², where M is the mass of the steel and R is the initial radius.

Final Situation:
In the final situation, the radius of the circle is halved, which means the new radius (R') is equal to R/2. Therefore, the new moment of inertia (I') of the steel mass is given by: I' = M(R/2)² = (1/4)MR².

Conservation of Angular Momentum Equation:
According to the conservation of angular momentum, the initial angular momentum (L) of the steel mass should be equal to the final angular momentum (L') after the change in radius.

Initial Angular Momentum: L = Iω = MR²(2ω) = 2MωR²

Final Angular Momentum: L' = I'ω' = (1/4)MR²ω'

Since the angular momentum is conserved, we can equate the initial and final angular momenta to find the new angular velocity (ω'):

2MωR² = (1/4)MR²ω'

Calculation of New Angular Velocity:
Simplifying the equation, we get:

ω' = 8ω

Therefore, the new angular velocity (ω') is 8 times the initial angular velocity (ω).

Conclusion:
When the radius of the circle is halved, the new angular velocity of the steel mass becomes 8 times the initial angular velocity. This change in angular velocity is a result of the conservation of angular momentum, where the moment of inertia decreases with the decrease in radius, leading to an increase in angular velocity to maintain the angular momentum.